How to determine if this function is one-to-one, onto, or bijection?

This notation means that the "x" in your function is a pair of integers $(m,n)$.

So the question is: Do you know two pairs $(m_1,n_1)$ and $(m_2,n_2)$ of integers that give the same $m_1^2+n_1=m_2^2+n_2$?

The two pairs count as distinct if at least one element changes.

1. one-to-one?

Choose two different $m$s and try to find $n$s such that the image of the function is the same for the two pairs.

1. onto?

Try $m=0$.